On 5/22/2012 10:56 AM, Joseph Knight wrote:
On Tue, May 22, 2012 at 7:36 AM, Stephen P. King
<stephe...@charter.net <mailto:stephe...@charter.net>> wrote:
On 5/21/2012 6:26 PM, Russell Standish wrote:
I once thought that consistency, in mathematics, was the
indication of existence but situations like this make that idea a
point of contention... CH and AoC
<http://en.wikipedia.org/wiki/Axiom_of_choice> are two axioms
associated with ZF set theory that have lead some people
(including me) to consider a wider interpretation of mathematics.
What if all possible consistent mathematical theories must somehow
Joel David Hamkins introduced the "set-theoretic multiverse" idea
(link <http://arxiv.org/abs/1108.4223>). The abstract reads:
"The multiverse view in set theory, introduced and argued for in this
article, is the view that there are many distinct concepts of set,
each instantiated in a corresponding set-theoretic universe. The
universe view, in contrast, asserts that there is an absolute
background set concept, with a corresponding absolute set-theoretic
universe in which every set-theoretic question has a definite answer.
The multiverse position, I argue, explains our experience with the
enormous diversity of set-theoretic possibilities, a phenomenon that
challenges the universe view. In particular, I argue that the
continuum hypothesis is settled on the multiverse view by our
extensive knowledge about how it behaves in the multiverse, and as a
result it can no longer be settled in the manner formerly hoped for."
Thank you for this comment and link! Do you think that there is a
possibility of an "invariance theory", like Special relativity but for
mathematics, at the end of this chain of reasoning? My thinking is that
any form of consciousness or theory of knowledge has to assume that
there is something meaningful to the idea that knowledge implies agency
<http://en.wikipedia.org/wiki/Agency_%28philosophy%29> and intention
Its one reason why Bruno would like to restrict ontology to machines,
or at most integers - echoing Kronecker's quotable "God made the
integers, all else is the work of man".
I understand that, but this choice to restrict makes Bruno's
Idealism even more perplexing to me; how is it that the Integers
are given such special status, especially when we cast aside all
possibility (within our ontology) of the "reality" of the physical
world? Without the physical world to act as a "selection"
mechanism for what is "Real", why the bias for integers? This has
been a question that I have tried to get answered to no avail.
I think Bruno gives such high status to the natural numbers because
they are perhaps the least-doubt-able mathematical entities there are.
The very fact that talks of a "set-theoretic multiverse" exist makes
one ask, how real are sets? Do set theories tell us more about our
minds than they do about the mathematical world? (Obviously, as David
Lewis pointed out, you need something like a set theory in order to do
mathematics at all, and as Russell says, for the average mathematician
it really doesn't matter.)
My skeptisism centers on the ambiguity of the metric that defines
"the least-doubt-able mathematical entities there are". We operate as if
there is a clear domain of meaning to this phrase and yet are free to
range outside it at will without self-contradiction. Set theory, whether
implicit of explicitly acknowledged seems to be a requirement for
communication of the 1st person content. Is it necessary for
consciousness itself? Might consciousness, boiled down to its essence,
be the act of making a distinction itself?
Also: *No one here has questioned the reality of the physical world.
*Should I append this statement to every email until you stop
I frankly have to explicitly mention this because the "reality of
the physical world" is, in fact, being questioned by many posters on
this list. That you would write this remark is puzzling to me. I think
that I can safely assume that you have read Bruno's papers... Maybe the
problem is that I fail to see how reducing the physical world to the
epiphenomena of numbers does not also remove its "reality".
Thank you for this clarification! Would you care to elaborate on
This is the origin of Bruno's claim that COMP entails that physics is
not computable, a corrolory of which is that Digital Physics is
refuted (since DP=>COMP).
Does the symbol "=>" mean "implies"? I get confused ...
Yes, that is the usual meaning. It can also be written (DP or not COMP).
"=>" = "or not"]
Actually "a implies b" is defined as "not a or b".
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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